Boundary conditions for constraint systems in variational principle

Abstract

1. Introduction

Constrained systems appear in many areas of physics, such as gauge theories and theories of gravity. For details see, e.g., Ref. [1]. Most of these theories are described through Lagrangians and their equations of motion are obtained through the variational principle, where boundary conditions are required to be imposed. Since the imposition of boundary conditions in the variational principle implies that the initial and/or final conditions are given in the equations of motion, the number of imposed boundary conditions in the variational principle should match the necessary and sufficient number of boundary conditions to obtain a solution from the equations of motion.

Despite their importance, the boundary conditions for the variational principle are only vaguely explored in theories with constraints. In Ref. [2], the action

was introduced as an example. A possible set of naive boundary conditions in the variational principle here might be δq₁ = δq₂ = 0 at the initial and final points, t = t₁, t₂. However, we should impose only one boundary condition at each boundary in the equation of motion as otherwise the system becomes over-constrained. This also implies that in the variational principle, to obtain the equation of motion, more than one boundary condition should not be imposed at each boundary; i.e., imposing boundary conditions δq₁ = δq₂ = 0 in the variation principle is inconsistent with the Dirac procedure. In fact, since the action (1) can be obtained from a point-particle action

by changing the variables |$q_3 \equiv \sqrt{m} q_1\pm \sqrt{M} q_2$|⁠, the authors concluded that the correct boundary conditions are δq₃(t₁) = δq₃(t₂) = 0. Reasonable boundary conditions can be imposed in this example because the “seed” action, from which the constraint systems originate, is obvious. For general cases, however, the systematical approach has not yet been explored. We emphasize that only after boundary conditions are adequately fixed can the necessary boundary (counter)terms be elucidated to make the variational principle well posed. This is the main topic that we address in this paper. For example, many gravity theories often introduce counterterms such as the Gibbons–Hawking–York counterterm by simply imposing the Dirichlet boundary conditions. However, this may encounter the same difficulties since these theories are also degenerate systems. Actually, some controversies have already arisen, e.g., whether a Gibbons–Hawking–York-type counterterm is necessary in Palatini theories of gravity [3–5], even though these theories have no second-order derivative in the action function. In order to address this kind of question adequately, we first should reconsider adequate boundary conditions in degenerate systems¹ that are consistent with the variational principle. This is what we would like to address in this paper.

The construction of this paper is as follows. We first review the boundary conditions for non-constrained/non-degenerate systems. We show that, for such systems, non-degeneracy directly implies that the number of necessary boundary conditions precisely coincides with the number of variables in the theory. Then in Sect. 3 we point out the mismatch of the number of boundary conditions and the number of variables for degenerate systems. For consistency in the variational principle, only the same number as the physical degrees of freedom, which is less than the number of variables in constrained systems, can be imposed. To construct the well-posed variational principle, we then outline systematic steps to obtain the correct number and form of boundary conditions. In Sect. 4, we demonstrate the proposed steps in some examples and show that well-posed boundary conditions in the variational principle are obtained. We also see in Sect. 5 that our procedures work well in general theory to some extent and explicit formulae for variables to be fixed and surface terms are given. Finally, in Sect. 6 we conclude our findings and expand on possible application to field theories.

2. Recap of boundary terms for unconstrained systems

First of all, let us review the boundary terms for unconstrained systems. Consider the following action of an N-point-particle system:

with i = 1, ⋅⋅⋅,N. The variation gives

Here we used the shorthand notation |$f_{x^i}=\frac{\partial f}{\partial x^i}$| and |$f_{\dot{x}^i}=\frac{\partial f}{\partial \dot{x}^i}$|⁠. In order to derive equations of motion and to make δS = 0, the boundary term is required to vanish. In the usual manipulation, the Dirichlet boundary conditions

are imposed. Then the variational principle gives equations of motion for the N point particles in a well-posed manner:

Hereinafter in this section, we consider a case in which the theory is unconstrained, which is characterized by

The N × N matrix |$f_{\dot{x}^i \dot{x}^j}$| is called the kinetic matrix.

Now we show that, if Eq. (7) holds, the minimum number of boundary conditions for a well-posed variational principle is at least 2N. This statement is proven by considering the contraposition. Hence, let us assume that the number of boundary conditions can be reduced to 2M( < 2N). This implies that the boundary term in Eq. (4) is written with functions |$F_a(x^i,\dot{x}^i)$| and X^a(xⁱ) where a runs from 1 to M( < N), i.e.,

Therefore, the well-posed variational principle requires the 2M boundary conditions to be

The relation (8) can be expressed as

Note that Eq. (11) holds for any δxⁱ, which gives

Taking the derivative with respect to |$\dot{x}^j$|⁠, we obtain,

Note that |$\frac{\partial F_a}{\partial \dot{x}^i}$| is an N × M rectangular matrix while |$\frac{\partial X^a}{\partial x^j}$| is an M × N rectangular matrix. Therefore, the determinant of the product of the two rectangular matrices is zero due to Cauchy–Binet’s theorem, i.e., |$\det f_{\dot{x}^i\dot{x}^j}=0$|⁠. The contraposition means the first statement of the paragraph.

Finally, let us confirm that the number of variables for this theory matches that of the dynamical degrees of freedom, which can be verified in Hamiltonian analysis. The canonical momentum of Eq. (3) is defined as

Note that if Eq. (7) holds, the inverse function theorem guarantees that Eq. (14) is solved for |$\dot{x}^i$|⁠, i.e., |$\dot{x}^i = \dot{x}^i(x^i,p_i)$|⁠. Then, the total Hamiltonian is simply written as

Thus, the phase space spans through all the canonical variables, namely (p_i, xⁱ) with i = 1, ⋅⋅⋅, N, which for real space indicates there are 2N dynamical degrees of freedom.

As a conclusion for this section, the number of boundary conditions for the variational principle coincides with that of the physical degrees of freedom in unconstrained systems, which is twice that of the physical degrees of freedom in the terminology of the Hamiltonian analysis.

3. Procedures for deriving well-posed boundary conditions

In general, boundary conditions for the variational principle in constrained systems are not as simple as their unconstrained counterparts. This is because the number of physical degrees of freedom, which is revealed from performing the Dirac procedure, does not coincide with the actual variables present in the theory. For example, the two point-particle Lagrangians |$L =\frac{1}{2}\dot{x}^2 +\frac{1}{2}\dot{y}^2$| and |$\tilde{L}=\frac{1}{2}(\dot{x}+\dot{y})^2$| may have the same number of variables but the Dirac procedure reveals that the former has two physical degrees of freedom while the latter has only one.

Through the equations of motion, boundary conditions determine a solution of a given theory. Too many boundary conditions generically give no solutions, and thus well-posed boundary conditions need to be imposed. Let us see this point from the viewpoint of the variational principle. We first fix boundary values for some variables. Then, the variation of the action gives equations of motion, if a sufficient number of boundary values are fixed such that the boundary contributions vanish. However, if too many boundary conditions are given by the variational principle, they become inconsistent with the equations of motion and the constraint structure that are obtained from the Dirac procedure. Although the final result might sometimes become consistent with too many boundary conditions, this is accidental and does not occur in general situations. Therefore, a sufficient number of boundary conditions for adequate variables is required such that the variational principle holds, but there should not be so many as to be inconsistent with the equations of motion and the constraint structure that are revealed by performing the Dirac procedure.

Thus, let us introduce the concept of well-posed boundary conditions for the variational principle as those satisfying the following: 1. The variational principle holds; i.e., the boundary contribution vanishes in the varied action; and 2. The minimal possible number of bQoundary conditions, which turns out to coincide with the number of physical degrees of freedom from performing the Dirac procedure, are imposed while still being consistent with the equations of motion/constraints and all of its solutions. We will show that the following five steps give well-posed boundary conditions for the variational principle in a given constrained/degenerate Lagrangian system.

• Find the primary constraint(s) and derive the total Hamiltonian.

• Find all other constraints (secondary, tertiary, etc).

• Construct coordinate and momentum variables |$(q^{a}_{\perp },p_{a\perp })$| that are orthogonal to the constraints with respect to the Poisson bracket.

• Convert all the Hamiltonian variables back to the Lagrangian variables.

• Rewrite the various boundary terms with respect to the orthogonal variables, up to surface terms, then apply all the constraints.

The existence of variables in Step 3 is guaranteed by a novel theorem proposed by S. Shanmugadhasan [7–9] and rediscovered by D. Dominici and J. Gomis and proved by T. Maskawa and H. Nakajima [10]. This theorem also states that the original coordinate variables of the given system can be transformed in a canonical manner, and guarantees that, in Steps 4 and 5, the boundary term is always rewritten in terms only of the variables. That is, there exists a canonical transformation

where Φ_α and the combination of Θ^μ and Θ_μ are the first-class and the second-class constraints, respectively, which are derived from performing the Dirac procedure. The surface term in the variation of an action can be written as

W is a surface term originating from the canonical transformation and ≈ is the weak equality in the Dirac procedure. To make the theory well posed, a counterterm −W has to be added to remove the surface term +W. Note that terms Φ_αδΞ^α and Θ_μδΘ^μ disappear not as a result of taking the variation to be zero, i.e., δΞ^α = δΘ^μ = 0, but because the coefficients become zero due to the constraints. Therefore, only the boundary condition |$\delta q^{a}_{\perp }=0$| needs to be imposed and then the variational principle becomes consistent with the equations of motion including the boundary terms with constraints. Then we naturally confirm the coincidence between the number of the physical degrees of freedom, which is none other than the half-number of the time-evolutive canonical variables |$(q^{a}_{\perp },p_{a\perp })$|⁠, and that of the boundary condition(s) for the variational principle.

In order to find pairs of canonical variables |$(q^{a}_{\perp },p_{a\perp })$| orthogonal to the constraints in Step 3, the following formula for an arbitrary function ϕ gives an insightful clue for cases only with the second-class constraints |$\theta _{\bar{\mu }}=0$|⁠:

where D is a Dirac matrix defined as |$D_{\bar{\mu }\bar{\nu }}:=\lbrace \theta _{\bar{\mu }},\theta _{\bar{\nu }}\rbrace$|⁠. We can check that the variable ϕ_⊥ is orthogonal to all the second-class constraints in the sense of the weak equality. Note, however, that this prescription does not provide canonical pairs but just gives some orthogonal variables. We need further manipulations for the construction of orthogonal canonical pairs, but unfortunately have no universal prescription. However, it is enough to recognize the necessity of additional boundary terms even for first-order derivative theories including the Palatini gravity, as shown in Sect. 5.

4. Examples of boundary conditions for constrained/degenerate systems

In this section, we shall go through the five steps in Sect. 3 in specific examples.

4.1. Two-particle system with a constraint

Let us first start with a simple example of a two-particle system with a constraint. The action considered here is

which has three variables (x, y, λ). Notice that the determinant of the kinetic matrix for (x, y, λ) is zero.

The constraint is obtained by varying with respect to λ:

This constraint relates the variables x and y, reducing the actual physical degrees of freedom, which we shall look into in more depth later in this subsection.

The variation of the action has boundary terms

For a well-posed variational principle, the boundary terms need to disappear, which implies

One may be tempted to immediately assume that

This implies that both x and y are fixed to arbitrary values on the boundary before the variation. However, this fixing in the variational principle generically becomes inconsistent with the constraint given in Eq. (21), except in accidental cases.

Let us turn to the Hamiltonian formalism, i.e., the Dirac procedure, to see the physical degrees of freedom clearly. The conjugate momenta are

We see that there is a primary constraint 0 = Φ ≔ p_λ. Thus, as Step 1, we write the total Hamiltonian as

In Step 2, all the constraints for this system need to be derived. The consistency condition for the primary constraint gives rise to a secondary constraint that then leads to tertiary and quaternary constraints as

The constraints (Φ, Ψ, Ξ, Θ) are all second-class and thus the physical degrees of freedom are (6 − 4)/2 = 1.

Step 3 requires us to find variables orthogonal to the constraints. Such “proper” coordinate and momentum variables can be taken as

which are indeed orthogonal to all the constraints. Step 4 is to rewrite the Hamiltonian variables into Lagrangian variables, which is achieved with the canonical relations of Eq. (26) as

Thus, in the final Step 5, the boundary term of the varied action (22) is rewritten as

Since Ξ = Φ = 0 is satisfied for any configuration of solutions including on the boundaries, only the boundary conditions δx_⊥(t₂) = δx_⊥(t₁) = 0 are sufficient for the variational principle to be well posed. We emphasize that the last two terms in Eq. (34) disappear not because of δΨ = δΘ = 0 but due to Ξ = Φ = 0. We do not impose any boundary conditions for δΨ and δΘ. Moreover, any fixing of x_⊥ on the boundary is consistent with the equations of motion and constraints.

4.2. Two-particle degenerate system: First- and second-class constraints

In the previous subsection, a constraint was introduced to a two-particle system through a Lagrangian multiplier. Here, we investigate a theory where an additional constraint arises due to the structure of the functional form of the Lagrangian. This is usually called “degeneracy”. As a simple example, we introduce an action

For the case with the + sign, the variable redefinition from x + y to q ≔ x + y transforms L_B into

Thus, this system has obviously one physical degree of freedom, and the corresponding boundary conditions are those for only q. In contrast, the case with the − sign is non-trivial and requires the five-step procedure to find well-posed boundary conditions for the well-posed variational principle.

The varied action has the following boundary term:

For the identification of the well-posed boundary condition, let us proceed to Step 1. The conjugate momenta are

Clearly, there is a primary constraint Φ ≔ p_x − p_y = 0. The total Hamiltonian is thus

As for Step 2, the consistency condition for the primary constraint Φ gives

We see that, for the + sign and for m = 0, the constraint Φ becomes a first-class constraint, whereas, for the − sign, Φ is a second-class constraint.

In all cases, the orthogonal variables, following Step 3, can be taken as

Note that, for all cases, there is a relation²

The next step, Step 4, of rewriting the variables and the constraints with respect to Lagrangian variables, gives

Thus, for the final Step 5, the boundary term of the varied action (38) is rewritten as

Since Φ = 0 for all configurations of solutions everywhere including the boundaries, δx_⊥(t₂) = δx_⊥(t₁) = 0 is sufficient for the variational principle to be well posed for this theory. Moreover, the number of conditions on each boundary coincides with that of the physical degrees of freedom.

4.3. One-particle degenerate system with second-order derivatives

The next example is a one-particle theory with second-order derivatives:

This is none other than a point-particle Lagrangian in disguise with a surface term

For the point-particle Lagrangian |$\tilde{L}_C$|⁠, one has to impose only two boundary conditions δx(t₂) = δx(t₁) = 0. The varied action of L_C, however, becomes

which at first glance needs four boundary conditions, |$\delta \dot{x}(t_2)=\delta \dot{x}(t_1)=\delta x(t_2)=\delta x(t_1)=0$|⁠; i.e., two more conditions are required in addition to those for |$\tilde{L}_C$|⁠.

With the above in mind, let us go through the five steps to identify the well-posed boundary conditions in the variational principle for the theory of L_C. Before that, however, let the order of the derivatives be reduced to

which is dynamically equivalent to L_C, as can be seen by solving |$\dot{x}=y$|⁠.³ Under variation, this action has the following boundary:

Now, let us start with Step 1. The conjugate momenta of |$L^\prime _C$| are

which gives three primary constraints:

Thus the total Hamiltonian becomes

In Step 2, the evolutions of the primary constraints become

which leads to a secondary constraint

All the constraints are second-class, which indicates that there is (6 − 4)/2 = 1 physical degree of freedom, as expected. As Step 3, the variables that are orthogonal to the constraints are taken as

Rewriting the constraints with respect to the Lagrangian variables, in Step 4, we obtain

For the final Step 5, the boundary term of the varied action (53) is rewritten as

The boundary term, at first glance, does not seem to disappear under the proper boundary conditions of δx_⊥(t₂) = δx_⊥(t₁) = 0. Therefore, the variational principle with the Lagrangian (52) does not work well because the boundary terms in the varied action do not vanish. However, since Step 5 is achieved up to surface terms, the final two terms can be eliminated by introducing the counterterm of

to the Lagrangian |$L^\prime _C$|⁠; i.e., the Lagrangian is deformed into

Then the newly introduced Lagrangian |$L^{\prime \prime }_C$| indeed has a well-posed variational principle with the boundary condition of δx_⊥(t₂) = δx_⊥(t₁) = 0. Explicitly, the new Lagrangian is of the form

This is none other than the point-particle Lagrangian |$\tilde{L}_C =\frac{1}{2} \dot{x}^2 - \frac{1}{2}m^2 x^2$| once the constraint |$y=\dot{x}$| is substituted. Thus, following the steps and obtaining the correct boundary conditions, one may obtain the counterterm that is necessary to make the variational principle of the theory well posed.

5. General cases

In the previous section, we demonstrated how our proposed procedure with five steps works in specific examples. Here, we will see that it works in the cases where the functional form is general to some extent.

5.1. Degenerate two-particle system: Purely kinetic case

Let us consider a two-particle K-essence-type Lagrangian,

which is assumed to be degenerate; i.e., |$0= K_{\dot{x}\dot{x}}K_{\dot{\phi }\dot{\phi }}-K_{\dot{x}\dot{\phi }}^2$|⁠. We assume that the degree of the degeneracy is unity; i.e., the matrix K_ij is not a null matrix. Without loss of generality, we consider a case where |$K_{\dot{x}\dot{x}}\ne 0$|⁠. Then, in particular, if |$K_{\dot{x} \dot{\phi }}=0$|⁠, then the variables x and ϕ are separated as |$L_{K}=f(\dot{x})+c\dot{\phi }$|⁠, where c is a constant, and the counterterm for the variable ϕ obviously becomes W = −cϕ. In order to avoid such trivial cases, we further impose a condition: |$K_{\dot{x} \dot{\phi }}\ne 0$|⁠. The surface term in the varied Lagrangian is

Similar to the examples in the previous section, for a degenerate theory, the imposition of the Dirichlet boundary condition for both x and ϕ does not give a well-posed variational principle.

The conjugate momenta are

Since the system is degenerate, there exists a primary constraint, which is written as

at least locally in (p, π) space. Taking a derivative of the momenta with respect to p, we obtain

The degeneracy condition gives us a relation between π and p:

The total Hamiltonian is then

with |$\dot{x}= \dot{x}(p)$| and |$\dot{\phi }=\dot{\phi }(p)$|⁠. The consistency condition of the constraint Φ shows that it is first-class:

Thus there are two dynamical degrees of freedom in phase space, i.e., one physical degree of freedom.

Let us search for a variable q_⊥ = q_⊥(x, ϕ, p) orthogonal to Φ, if it exists. Such a variable q_⊥ is required to satisfy the partial differential equation

By taking a variable q_⊥ such that it has a structure

we see that q_⊥ is always orthogonal to Φ. The conjugate momentum for this variable is

with |$q^\prime _\perp =\partial _\phi q_\perp$|⁠. Taking |$q_\perp =\phi +\left(\frac{\partial F}{\partial p}\right)^{-1}x$| and |$p_\perp =\frac{1}{2}(\pi +F)$|⁠, the surface term is rewritten as

The variable that should be fixed on the boundary is only |$\phi +\frac{K_{\dot{x}\dot{x}}}{K_{\dot{x}\dot{\phi }}}x$|⁠. The counterterm of

needs to be added to the Lagrangian L_K, for the well-posedness of the variational principle. Note that, even though the Lagrangian L_K has only first-order derivatives, additional surface terms are required.

5.2. Degenerate two-particle system: Separated kinetic and potential terms

Let us consider a two-particle system in which the kinetic and potential terms are separated; i.e., the Lagrangian is given as

The surface term of this Lagrangian is

We assume that the kinetic term is degenerate but non-zero. Without loss of generality, we can impose conditions for the kinetic term as

The conjugate momenta for x and ϕ become

respectively. Since the kinetic matrix is degenerate, p and π are related, which gives a primary constraint. We write the primary constraint as

which can be justified by |$K_{\dot{x}\dot{x}} \ne 0$|⁠. The total Hamiltonian becomes

We define Φ as

and then the primary constraint (86) is expressed as

The time evolution of the primary constraint Φ should be zero, which gives a secondary constraint:

We will show the relation between F and K. The primary constraint (89) should be an identity, if we write it in terms of |$\dot{x}$| and |$\dot{\phi }$|⁠. Hence, we have

Its variation gives

i.e.,

The above two equations are the same because of Eq. (84), and give

The Poisson bracket of Φ and G becomes

Suppose that {Φ, G} is non-zero for the system and has one physical degree of freedom.

The orthogonal component of a physical variable f to constraints Φ and G is written as

The normal component of p is written as

and, substituting Eq. (85) (note that then Φ = 0), we have

where |$\hat{=}$| means that we substitute Eq. (85). This may mean that p would be a good variable for a momentum of the physical degree of freedom. The normal component of x becomes

Since x_⊥ has a term proportional to G, the Poisson bracket of x_⊥ and p_⊥ does not give unity, which means that a pair of x_⊥ and p_⊥ is not conjugate. It may be interesting to consider a variable |$x+ {\cal F}(p) \phi$|⁠, which is a conjugate partner of p in (x, ϕ, p, π) space and corresponds to |$\phi +\frac{K_{\dot{x}\dot{x}}}{K_{\dot{x}\dot{\phi }}}x=\frac{K_{\dot{x}\dot{x}}}{K_{\dot{x}\dot{\phi }}}(x+\frac{K_{\dot{x}\dot{\phi }}}{K_{\dot{x}\dot{x}}}\phi )$| in Eq. (80) for the case with V = 0. The normal component of |$x+ {\cal F}(p) \phi$| is

Therefore, if |${\cal F}= F_p$|⁠, the above becomes

which might be a good conjugate pair of p_⊥. Actually, the Poisson bracket of (x + F_pϕ)_⊥ and p_⊥ gives unity after substitution of Eq. (85). Then, the surface term of δL_V is expected to be written as pδ(x + F_pϕ).

Let us check that the surface term of δL_V becomes pδ(x + F_pϕ). The surface term of δL_V is

Then, adding the counterterm of

the original surface term of the varied action becomes

This is consistent with the result obtained in the Hamiltonian analysis. Note again that, in order to make the variational principle well posed, i.e., compatible with the Dirac procedure, additional surface terms are required, even though the Lagrangian L_V has only first-order derivatives.

6. Conclusion

In this paper, the methodology of obtaining well-posed boundary conditions for the variational principle in constrained/degenerate systems is presented. First, we reviewed the boundary conditions for non-constrained systems. It was noted that the number of variables, that of the physical degrees of freedom and that of the conditions imposed on each boundary in the variational principle coincide, with one another for such a system. On the other hand, it was then pointed out that for constraint systems such relations sometimes do not hold and the number of variables does not match the number of physical variables. This makes a nonsense of the variational principle; i.e., having too many constraints on the boundary in the variational principle leads to inconsistency with the Dirac procedure. Thus, we proposed a five-step procedure to give well-posed boundary conditions for the variational principle in a given constrained/degenerate Lagrangian system. We then proceeded to use these methods to derive the boundary conditions for some specific examples of multi-particle and higher-order derivative theories. Finally, we showed that it works in cases with a general functional form and gave explicit formulae for the variables to be fixed as well as the surface terms. We emphasize that, even in a theory without higher-order derivatives such as the Palatini formalism of general relativity, surface terms need to be introduced for the well-posed variational principle.

In the future, extension to field theories can be considered. For instance, general relativity is a degenerate theory in the sense that it has second-order derivatives in the action but its equations of motion remain of second order. As is well known, in order to obtain the proper (Dirichlet) boundary term, one has to introduce the Gibbons–Hawking–York counterterm, which was proposed first in Ref. [12] and then by Gibbons and Hawking in Ref. [13]. This has been discussed for some time [3,14–19]. The derivation of the Gibbons–Hawking–York term, despite its importance, consists of guessing the functional form of the counterterm under the assumptions that the Dirichlet boundary conditions are imposed on boundaries and that the spatial diffeomorphism [19] is respected. However, we revealed that simple Dirichlet boundary conditions are not necessarily adequate in degenerate systems. This implies that systematic, rather than heuristic, derivations of counterterms including the well-known Gibbons–Hawking–York counterterm should be performed under the appropriate boundary conditions derived by the five-step procedure in Sect. 3. Although the simple Dirichlet boundary condition miraculously works well for general relativity, it may not for gravitational theories with more constraints, such as Palatini theories of gravity [4,5]. These arguments will be discussed elsewhere in the future.

Funding

Open Access funding: SCOAP³.

Acknowledgement

We would like to deeply thank Mu-In Park for his insightful and pointed question that initiated this work. K.S. would also like to thank Katsuki Aoki, Shoichiro Miyashita, and Alexander Vikman for their fruitful and helpful discussions. K.I. is supported by JSPS Grants-in-Aid for Scientific Research Numbers JP17H01091, JP21H05182, JP21H05189, JP20H01902, and JSPS Bilateral Joint Research Projects (JSPS-DST collaboration) for Scientific Research Number JPJSBP120227705. K.S. was supported by a JSPS KAKENHI Grant Number JP20J12585 during the initial stages of this work. K.T. is supported by a Tokyo Tech Fund Hidetoshi Kusama Scholarship. M.Y. is supported by IBS under the project code IBS-R018-D3 and by a JSPS Grant-in-Aid for Scientific Research Number JP21H01080.

Footnotes

References